NOTES 



DRAWING 




Class. 
Book. 



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COPYRIGHT DEFOSm 



Notes on Drawing 

AS APPLIED TO THE 

COURSE IN VEHICLE DRAFTING AND 
CONSTRUCTION 

OF 

THE CARRIAGE MONTHLY 



By 

EDWARD E. KRAUSS 

B.S. in M.E., University of Pennsylvania 

Instructor in Vehicle Drafting and Mechanical Drawing 

Central Educational Institute, Philadelphia, Pa. 



PUBLISHED BY 

\?V>^RE: BROS. C01VIF>ANY 

PHILADELPHIA. U. S. A. 
1914 



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^HIS BOOK forms part of The 
-■- Carriage Monthly Course in 
Vehicle Drafting and Construction, 
prepared by Chas. A. Heergeist, tech- 
nical editor of The Carriage MonthIvY. 



Copyrighted 1914 
Ware Bros. Company 

FEB 10 1914 



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©CI.A3625 24 



PREFACE 

THERE has no doubt ofttimes been apparent to teachers 
of mechanical drawing the need of a booklet giving 
instructions for the handling of instruments, general 
data for drawing and a summary of geometrical definitions, 
which could be placed in the hands of beginners. The 
present ''Notes on Drawing" have been prepared, based on 
experience in teaching apprentices and mechanics, in an 
endeavor to supply this need especially in connection with 
The Carriage Monthly Course in Vehicle Drafting and 
Construction, prepared by Chas. A. Heergeist, Technical 
Editor of The Carriage Monthly. 

It is desired to make acknowledgment to Wm. C. H. 
Slagle, Assistant Professor of Drawing, University of Penn- 
sylvania, for helpful suggestions on drawing ; to Dr. George 
H. Hallett, Professor of Mathematics, University of Penn- 
sylvania, for reading the manuscript on Geometrical Defi- 
nitions ; to Chas. A. Heergeist, Technical Editor of The 
Carriage Monthly, for the preparation of the drawings ; 
and to F. Weber & Company, for the use of some of the 

illustrations. ^ ^ ^^ 

E. E. K. 

Philadelphia, Pa., January, 1914. 




Fig. I. Drawing Instruments. 



Fig. 2. Drawing Board. 



INSTRUCTIONS FOR MAKING DRAWINGS 

A drawing is the representation of objects on a plane surface by 
means of lines. Tiiere are "freehand drawings" which are made 
entirely by hand and "mechanical drawings", made with the aid of 
instruments. Vehicle drawing is a branch of mechanical drawing. 

The Bssentials. In making a mechanical drawing the essentials 
are accuracy, clearness and neatness, as the object of such a drawing 
is to enable one to make the part or combination pictured, without 
any other information than that contained on the drawing. The 
workman constructs as the drawing shows, not as the draftsman 
may have intended to show. 

I. Instruments and Their Use 

The following instructions should be carefully studied and kept 
in mind when making drawings : 

I. List of Instruments and Materials. 

a. Drawing board 22 x 30 inches. 

b. Set of drawing instruments in a case, Fig. i, containing — 

Compass, 4^ in., with pencil and pen attachments, 

lengthening bar and extra needle point. 
Bow pencil, 3^ in. 
Bow pen, 3^ in. 
Ruling pen, 5 in., with cleaning device. 

c. Triangular boxw^ood scale, having scales of 4, 3, 2, i^, 

i^, I, j4 and Yz inches to the foot and 50 parts to the 
inch, divided the full length of the scale. 

d. T-square, 30 in. long. 

e. 45 degree celluloid triangle, 7-in. 
/. 60 degree celluloid triangle, 9-in. 
g. Irregular curve, No. 16, celluloid. 
h. Irregular curve, No. 20, celluloid. 

i. Two 6-H Koh-i-noor drawing pencils. 
y. Emerald pencil eraser. 
k. Cleaning rubber or artgum. 
/. ^ dozen thumb tacks. 
m. Drawing paper, 22 x 30 in. 
n. Lettering paper. 

o. Penholder with cork tip and Falcon pen. 
2. Drawing Board. The drawing board, at least 22 in. wide and 
30 in. long, should be made of strips of well-seasoned pine, about 

(5) 



6 Notes on Drawing. 

I in. thick, glued together (Fig. 2). Across the back there should 
be a series of grooves, ^2 in. deep, running with the grain of the 
wood to prevent warpiiig. There should be two ledges of hardwood 
across the back, held in place by screws in oval slots, to allow for 
contraction and expansion. 

The sides of the board should be perfectly straight and square. 

3. Drawing Paper. The drawing paper for this course should 
be cold pressed and smooth. The size is 22 x 30 in., which sheet 
cut in half will give two sheets 22 x 15 in. 




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Fig. 3. Method of Sharpening Pencil. A. Conical Point. B. Chisel Point. 

To fasten the paper to the board place a thumb tack in the upper 
left-hand corner of sheet and see that the upper edge of paper is in 
line with the T-square. Passing the left hand diagonally over the 
paper from the thumb tack to the lower right-hand corner, thus 
pulling the paper taut, place a thumb tack in the latter corner. Now 
fasten the other corners, pulling the paper taut from the center. 

4. Pencils. In order that fine lines may be drawn the lead of 
the pencil should be hard, 6-H hexagonal generally being used. With 
such a pencil very fine and clear lines can be made, from which 
more accurate measurements can be secured than if a softer pencil 
is used, and, in addition, it lasts longer. 

In drawing lines with a pencil always move the hand in a direc- 
tion away from the body, lettinsf the pencil lean in the direction 
the hand moves (Fig. 14). The pencil should always be drawn over 
the paper, not pushed. 

Sharpening of Pencils. One pencil should be sharpened to a con- 
ical or round point (A) for laying off, and the other to a chisel 
point (B) (Fig. 3), for drawing lines. To keep the pencil sharp 
rub the lead on a line file or a piece of fine emery cloth or sand- 
paper tacked to a little block of wood. 




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Use of Drawing Instruments. 9 

5. T-Square. The T-square (Fig. 4) consists of a long, light 
blade attached at right angles to a head which slides along the left 
side of the drawing board. All horizontal lines are drawn by mov- 
ing the pencil along the top edge of the T-square, whose head is 
held snug against the left edge of the board. Do not hold the head 
against the bottom edge of board to draw vertical lines. 

6. Triangles. The best triangles are made of transparent cellu- 
loid, not less than tV in. thick. Triangles of 45 and 60 degrees are 
generally used. (See Fig. 5). 





Fig. 9. Irregular Curves. 

All vertical or slanting lines are drawn with the aid of triangle 
(Fig. 4). 

To draw a line perpendicular to a slanting line. In Fig. 6 place 
the short edge of triangle A along line a b, perpendicular to which 
a line is to be drawn. Holding triangle A in place lay triangle B 
against it; slide triangle A to position A' and draw line c d, which 
will be perpendicular to line a b. 

The positions of triangles to obtain lines at various angles are 
shown in Fig. 7. To obtain other angles a protractor is used. 
(Fig. 8). 



10 Notes on Drawing. 

7. Irregular Curves or Sweeps. These are used for drawing 
smooth curves which are not parts of a circle, using either pencil or 
ruling pen. The two types of curves most useful are shown in Fig. 9. 

The proper method of using such curves is as follows : Suppose, 
in Fig. 10, it is desired to connect points a, b, c, d, e, and / by a 
smooth curve. Lay the irregular curve in position A so that one of 
its sweeps touches 3 points, say a, b and c, and draw the curve as 




Fig. 10. Method of Using Irregular Curves. 



far as c. Now start with b and cover b, c and d. Repeat until the 
curve is completed, always, however, starting back one point so as 
to obtain a smooth curve. 

8. Compass. The compass (Fig. 11) is used to draw arcs and 
circles, either in pencil or ink. Each leg is provided with a joint sa 
that the lower portion can be kept perpendicular and both nibs of 
the pen bear equally on the paper. The pin point should have a 
square shoulder so that it cannot make a large hole in the paper.- 



Use of Drawing Instruments. 



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11 *" 13 

Fig. II. Compass with Pen and Pencil Attachments and I^engthening Bar. 

Fig. 12. Bow-pencil and Bow-pen. 

Fig. 13. Ruling Fen. 



12 



Notes on Drawing. 





Fig. 14. Method of Using Ruling Pen. 



Use of Drawing Instruments. 13 

Always use a hard lead in the pencil attachment. 

When it is desired to draw large circles, the lengthening bar is 
inserted between the upper part of leg and the pencil or pen attach- 
ment. 

The compass should be gauged with one hand, the thumb and 
fourth finger holding the needle point leg, which is generally set on 
the drawing first, and the pencil or pen point between the first and 
second fingers, which move the point to the desired setting. 

When drawing circles the compass should be held at the top 
between the thumb and first finger, bringing slight pressure to 
bear on the pencil or pen point, but none on the pin point. The 
compass should lean slightly in the direction in which it rotates. 

9. Dividers. The dividers are simply a compass fitted with two 
long needle points and are used for transferring dimensions or 
spacing ofY. The compass shown in Fig. 11 can be converted into 
dividers by reversing the shoulder needle point and replacing the 
lead in the pencil attachment with the extra needle point provided. 
In using the dividers prick the paper only slightly; the smaller the 
mark the better. 

10. Bow^ Pencil and Bow Pen. (Fig. 12). These are used for 
drawing small arcs and circles, as they are much easier to handle 
and give better results. The needle point should be adjusted to the 
same length as the pencil or pen point. 

11. Ruling Pen. The ruling pen (Fig. 13) is used to ink in all 
straight lines and irregular curved lines. The pen should be held 
lightly in the right hand (Fig. 14) in a vertical position, so that 
both nibs touch the paper and inclined in the direction in which the 
pen moves. The third and fourth fingers should rest on and slide 
along the T-square or triangle. The pen should be held lightly and 
not pressed against the side of the T-square or triangle, the thumb 
screw being kept on the far side of the guiding edge. 

Ink is filled in the space between the nibs, not more than ^ in. 
along the blades, by means of the quill attached to the cork of the 
drawing ink bottle. Great care should be taken to keep the blades 
clean, and no ink should be on the outside, as blots will surely result. 
If the ink does not run freely, clean the pen by releasing the clean- 
ing device under the thumb screw and wiping the nibs with a 
soft rag. 

To adjust the ruling or compass pen for thickness of line, the 
thumb screw is rotated by the thumb and middle finger. Do not 



14 Notes on Drawing. 

try to draw so fine a line that the nibs of the pen touch each other, 
as the ink then can not flow freely. 

The pens cannot be cleaned too often. 

12. Erasers. For pencil line use the emerald eraser. For ink 
lines use the same eraser, and under no conditions use a knife, as 
this is bound to spoil the paper. For a general cleaning of the draw- 
ing without removing lines, use artgum or a soft cleaning rubber. 



II. Standards for Drawings 

1. Sizes. Every drawing room has standard sizes for the draw- 
ings so they can be filed and each easily located. In this course the 
sizes will be 14 x 20 in. and 20 x 28 in., with a ^ in. margin. The 
layout of a 14 X 20 in. sheet is given in Lesson i — Course A. 

2. Titles. Unless otherwise indicated, all drawings in this 
•course shall have titles as shown in Lesson i — Course A, neatly let- 
tered on the margin and reading from the outside, as shown. 




^5^ S^ 

Fig. 15. Arrangement of Views. 



3. Arrangement of Views. The different views of an object 
should be located as per Fig. 15. 

4. Lines. Each of the different kinds of lines (Fig. 16) used in 
a drawing has a definite meaning and use. All lines showing any 
part of the object, whether full, dotted or irregular lines, are of the 
same weight or thickness, except shade lines, which are three times 
as heavy ; all other lines are lighter. 

(IS) 



i6 



Notes on Drawing. 



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Fig. 1 6. Sample I^ines Showing Exact Weight to be Used on Drawings^ 



Standards of Drawings. 17 

a. Full Hues are used to show all unobstructed edges of the 
object. 

b. Dotted lines are used to show all hidden edges of the object. 
The first and last dot of a dotted line should touch the lines at which 
the hidden edge actual-ly terminates. 

c. Irregular lilies represent the edge of a surface where a part is 
broken away to show the interior. 

d. Center lines give the axis of symmetry, centers of circles, etc. 

e. Dimension lines are used to show the size of the various parts 
of an object. The arrowhead should be made with the stroke of the 
pen towards the point, which should just touch the line from where 
the dimension reads. 

/. Construction lines are used to show the method employed to 
obtain the result shown in the drawing. These should touch the 
points between which they are drawni. 

5. Inking In. A drawing should not be inked in until it is com- 
plete in pencil. The ink lines should be of the thickness shown in 
Fig. 16. 

In order to secure the best results and save time, the inking should 
be done in the following order : 

a. Ink all full or dotted line circles and circular arcs, beginning 
with the smallest. 

b. Ink all other curved lines, full or dotted. 

c. Ink all horizontal full and dotted lines, working downward 
from the top. 

d. Ink all vertical and inclined full and dotted lines, working 
from left to right. 

e. Ink all construction line circles and circular arcs. 

/. Ink all horizontal construction lines, working down from the 
top. 

g. Ink all vertical and inclined construction lines, from left to 
right. 

h. Ink all center and dimension lines. 

i. Ink cross hatching. 

y. Ink border lines, which should be heavier than the full lines. 

k. Ink dimension figures, notes and titles. 



III. Lettering 

To make a drawing entirely clear it is always necessary to place 
on the same dimensions, explanatory notes, titles, etc. This should 
be done with a plain, legible style of letter, easily executed. The 
appearance of a drawing, well made as far as the lines are con- 
cerned, is often spoiled by poor lettering. The beginner will gen- 
erally experience difficulty in executing neat lettering, but a little 
practice will produce wonderful results. 

1. Style. The style of lettering which seems to best meet prac- 
tical conditions is that developed by Mr. C. W. Reinhardt and given 
in Figs. 17 and 18. Slanting lettering is mostly used, although the 
same style can be executed vertical, which is done for all titles of 
large letters. Letters consist of capital letters (called upper case) 
and small letters (called lower case). 

2. Size. In general, the lower case or small letters are three- 
fifths the height of capital letters. The usual height is 5/32 inch 
for capitals, making the lower case letters Z^Z'^ inch, as shown in 
Fig. 17. Always draw parallel lines as a guide for top and bottom of 
letters as shown. This is necessary for beginners, and is done by 
experienced draftsmen. 

3. Slant. The slant or inclination of letters is usually i to 2^, 
about 70 degrees, as shown. An angle of 60 degrees is satisfactory, 
which enables one to draw the slanting guide lines by means of a 
6o-degree triangle held against the T-square. Such guide lines are 
necessary for a beginner and may be dispensed with only after suffi- 
cient practice enables one to give to all lettering a uniform inclina- 
tion which is essential to a good appearance. 

4. Execution. All lettering on drawings should first be penciled 
and then inked in. All practice lettering should be done with draw- 
ing ink. 

a. Pen'Point. A Falcon pen point is recommended, which should 
be limbered up so as not to be too stiflf. This is done by holding the 
nibs of the pen with a cloth and working the end of pen back and 
forth. One can then produce heavy lines without pressing on the 
pen unduly. 

b. Thickness of Lines. All the lines forming a letter should be 
of a uniform thickness. There should be no fine lines in any letter. 
They should all be as heavy as shown. 

(18) 



Lettering. 19 

c. Letters. The small and capital letters should be made exactly 
the same shape as shown. Following the lines showing the small 
letters, also the capitals, the same are repeated with the separate 
strokes of the pen shown, an arrow indicating the direction in which 
the pen moves in forming that part of the letter. It should be care- 
fully noted which part of the letter follows the general slant. 

d. Figures. The construction of the figures is shown the same 
as of the letters. In the case of fractions, a horizontal line, half way 
between top and bottom of regular figures, is drawn as the frac- 
tional dividing line and the figures of the fraction, Ys the size of the 
regular figures, are so made that a slant guide line through the cen- 
ter of the numerator will pass through the center of the denom- 
inator. 

5. Spacing of Letters and Words. The blank spaces between 
letters of the same word should always be the same, say about ^ 
the height of body of letter. For small letters this would be about 
1/32 inch, for capitals about 2/32 inch. 

The spaces between words should all be equal and about the 
length of the letter 1 or 5/32 inch. 

6. Practice Plates. The following practice plates should be 
carefully and neatly lettered, and on the completion of each individual 
letter the same carefully compared with the sample letters shown, 
so as to correct any discrepancies in the next letter. It is essential 
that the pen be moved very slowly in forming a letter, the hand con- 
stantly guiding the point to produce the proper shape. Especial 
care should be taken to extend letters fully to top and bottom guide 
lines ; the letters should not fall short or extend beyond. 

In the following plates one line of each letter given should be 
made and this repeated until the sheet of paper is filled : 

Plate I. i, 1, j, t, f 

II. h, m, n, u, y, v, w 

III. k, r, X, z, s, e 

IV. o, p, b, c, a, d, g, q 
V. I, L, F, E, H 

VI. A, V, W, K, X, Z 

VII. T, N, M, Y, J 

VIII. P, R, D, B, U 

IX. C, G, O, Q, S 

X. I, 7, 4, 3, 2, 5, o, 6, 9, 8. 



20 



Notes on Drawing. 



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I V. Sections and Cross-Hatching 

I. Sections. In order to make the construction of an object more 
clear, it is often desirable to show the interior. This is accom- 
plished by means of a section, i.e. the near portion of the object is 
considered to be cut away and the lines drawn to show the interior, 
Fig. 19. 

If a section is shown along a center line this should be indicated 
bv a note on the drawing "Section on A-B." If the section is not 




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Fig. 19. Drawing showing Sections and Cross-hatching. 

along a center line the edge of the near portion cut awa}^ is indicated 
by an irregular line. 

2. Cross-Hatching. In a section it is often necessary to cut 
through a portion of material, the character of which is indicated by 
the spacing and thickness of lines as per Fig. 20. The slant is 45 
degrees, unless otherwise indicated. 

(22) 



Sections and Cross-Hatching. 



23 



When two separate pieces, either of the same or different 
materials, adjoin each other, the cross-hatching lines are drawn in 
opposite directions. 




Cast Iron 




Cast Steel 




Brass 




Aluminum 




Leather 






Wood 





Wrought Iron Malleable Iron 




Wrought Steel 



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Bronze 




Bearing Metal 




Asbestos 



Liquid 




Tool Steel 




Copper 




Vulcanite 




Wires 




Glass 



Fig. 20. Conventional Cross-hatchings. 

The Spacing of the lines is done by the eye and uniformity is 
necessary to good results. It is often a great help to glance at the 
two or three previous spaces when setting the next one. 



24 



Notes on Drawing. 



3. Breaks. When a long piece cannot be shown full length and it 
is desired to show both ends, a break is made at a convenient place, 
as shown in Fig. 21. 



Cylindrical 




Wood 



Rectangular 



K 



Hollow Cylindrical 



Fig. 21. Method of showing Breaks in lyong Pieces. 



V. Scales 

1. Systems of Measurement. There are two systems of meas- 
urement of length used in drawings. These are the English or 
American, where the. unit of length is i foot, which is divided into 
twelve inches, and the French or Metric, where the unit of length 
is I meter, which is divided into ico centimeters or i,ooo millimeters. 

2. Scales of Drawings. It is not practical generally to lay out 
an object to actual dimensions on a sheet of paper of convenient 
size. Therefore, the object is drawn smaller, say ^, ^ or 1-16 of 
the actual size. In order to save time, use is made of a scale, such 
as shown in Fig. 22, where the lengths are shown to the reduced 
size. The triangular scale shown has six faces and different scales 
are marked on each of them. 




Fig. 22. Triangular Boxwood Scale. 

Drawings of vehicle bodies are sometimes made full size, but 
mostly to a reduced scale. 

In the American System, Fig. 23, for a scale of >4 inch to the 
foot, each i foot of actual length of the object is represented in the 
drawing by a length of ^ inch, which is one division of the scale. 
Each division is divided into twelve parts, each part representing 
I inch of actual length of the object. If the scale is i inch to the 
foot, then each i foot of length of the object is i inch long in the 
drawing, etc. 

In the Metric System, Fig. 24, for a scale of 5 centimeters to the 
meter, each i meter of actual length of the object is represented in 
the drawing by a length of 5 centimeters, which is one division of 
the scale. Here, however, each division is divided into ten parts, 
each part representing i-io of i meter or 10 centimeters of actual 
length of the object. 

(25) 



26 



Notes on Drawing. 







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3. Scales in General Use. The 
scales generally used are : 

American System — 14, yi, i, i34> ^ 
and 3 inches to the foot. 

Metric System.— 2y2, 3, 5 <^r ^^ 
centimeters to the meter. Small 
"Fashion Plates" are drawn to a 2^, 
or 3-centimeter scale, and large 
"Fashion Plates'' to a 5-centimeter 
scale. A lo-centimeter scale is 
mostly used for working drafts, and 
this represents a reduction of i to 10. 

4. Use of Scales. It will be noted 
that at the end of each face of scale^ 
Fig. 22, I ft. is divided into inches 
and fractions, and on the other side 
of the zero mark only feet divisions 
are indicated. To lay off a dimen- 
sion of feet and inches, for example 
2 ft. 3 in., measure from the 2 ft. 
mark back to o and then to the 3 in. 
mark. 

In laying off dimensions the scale 
is laid directly on the drawing and 
the desired length marked off with 
a sharp pencil. The divider or com- 
pass should not be held against the 
scale and the lengths then marked 
on the drawing, as this will not give 
accurate results and at the same time 
destroys the markings on the scale. 






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VI. Shading 

1. Shade: Lines. It is often desirable to make more clear the 
shape of an object by indicating the raised or depressed surfaces by 
means of shade lines and thus overcoming the flatness of a drawing. 
Shade lines are three times the thickness of full lines, see Fig. i6. 

It is considered that the light emanates from the upper left-hand 
corner of a drawing in parallel rays at an angle of 45 degrees, see 
Fig. 25. The shade line is used to separate a surface receiving the 
light from one which does not. It is not considered that one portion 
of an object casts shadows on another portion. By sliding a 45 
degree triangle along the T-square and assuming the hypotenuse 
to be a ray of light, the surfaces upon which the light rays do not 
impinge can be easily determined, and these then shaded. 

To shade a circle or arc draw a line at 45 degrees through the 
center and move the needle point, without changing the radius or 
the setting of pen, downward along this line a distance equal to the 
maximum thickness of shade line. The space between can be filled 
in by slightly springing the pen point. 

2. Line Shading. In order to more clearly illustrate the nature 
and relative positions of the surfaces of an object, line shading is 
resorted to. This is accomplished by drawing parallel or converging 
lines which decrease in thickness and increase in spacing from the 
darker to the lighter portion of a surface. The rays of light are 
considered to be at 45 degrees. 

The method of line shading various surfaces is illustrated in the 
following surfaces : 

Fig. 26. Cylinder and interior of curved surface. 



Fig. 27. 


Prism. 


Fig. 28. 


Cone. 


Fig. 29. 


Sphere. 


Fig. 30. 


Pyramid. 


Fig. 31. 


Combination of curved surfaces, 



3. Tint Shading. Shading by means of hand or air brush gives 
a much better effect and is used when a high class drawing is 
desired. Figs. 32-36 illustrate such work. 

(28) 



Shading. 



29 




Fig. 25. Shade I^ines. 



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Notes on Drawing. 





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Fig. 26. Method of Ivine Shading. 



Shading. 



31 






Figs. 27-30. Line Shading of Solid Bodies. 



32 



Notes on Drawing. 




Fig. 31. Method of lyine Shading Combination of Curved Surfaces. 



Shading. 



33 








Fig. ^2. Tint-shading. 



34 



Notes on Drawing. 







Fig. 33- 



Shading. 



35 








Fig. 34- 



36 



Notes on Drawing. 







Fig. 35. 



Shadinc- 



37 






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Fig. 30- 



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Notes on Drawing. 



LINES 



Point 



Straight 



Curved 



I I 

Convex. 



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Concave-Convex 



Broken 



Horizontal 




Vertical Perp0nd±cular to AB 



Oblique 



Oblique to AB Parallel 




Divergent 



Convergent 



Point of Intersection 



GEOMETRICAL DEFINITIONS 

Geometry is that branch of mathematics which treats of the prop- 
erties of hues, angles, surfaces and soHds. 

I. The Point 

I. A ^oint indicates position only. It has neither length, breadth 
nor thickness. 

II. The Line 

1. A I'uie has only one dimension, that is length. A line is con- 
sidered to be the path of a moving point. 

2. A straight line is one which does not change its direction in 
any part of its length. It is the shortest distance between two points. 

3. A curved line changes its direction at every point. 

a. A concave curved line is one which curves inwardly. 
h. A convex curved line is one which curves outwardly. 
c. A concave-convex or reverse curve line is one which has 
both a concave and a convex curve. 

3. A broken line consists of several joining straight lines of 
different directions. 

4. A horizontal line is one which is level with the horizon or with 
the surface of water at rest. A line parallel with the top of a draw- 
ing is generally called a horizontal line. 

5. A vertical line has the direction of a plumb bob line. It is at 
right angles to a horizontal line. 

6. A perpendicular line is a line vertical to a horizontal line. A 
line is perpendicular to another line when it does not incline on 
either side towards it. 

7. An oblique line is one which is neither horizontal nor vertical. 
A line is oblique to another line when it is not perpendicular to the 
line considered. 

8. Parallel lines are equi-distant from each other at every point. 

9. Divergent lines continually recede from each other, Such as 
lines radiating from one point. 

10. Convergent lines are continually coming closer to each other. 
These are also called contracting lines. 

11. The point of intersection is w^here two or more lines cross 
each other. 

(39) 



40 



Notes on Drawing. 



ANGLES 






straight Line 



Curved Line 



Spherioal. 



90 75, 




Right 




Acute 




Obtuse 



Adjacent 



Opposite 



Dihedral 




complementary 



Supplementary 



Geometrical Definitions. 41 

III. The Angle 

1. An angle is formed by two intersecting lines called the sides, 
and is measured in degrees. The point of intersection is called the 
vertex. 

2. A straight line angle is one in which the intersecting lines are 
straight lines. 

3. A curved line angle is one in which the intersecting lines are 
curves. The angle is measured by lines tangent to the curves at 
the vertex. 

4. A spherical angle is one in which the intersecting lines are 
arcs of the same radius. 

5. A degree is the 1/360 part of the circumference of a circle. 
Each degree is divided into 60 minutes, and each minute into 60 
seconds. 

6. A right angle is one in which the intersecting lines are perpen- 
dicular to each other. A right angle has 90 degrees, or one-quarter 
of a circle. 

7. An acute angle is smaller than a right angle and, therefore, 
has less than 90 degrees. 

8. An obtuse angle is greater than a right angle and has. there- 
fore, more than 90 degrees. 

9. An oblique angle is one which is not a right angle. 

10. Adjacent angles are angles which have one side and the 
vertex in common. 

11. Opposite or vertical angles are formed by two intersecting 
straight lines which form the two sides of the angles and the point 
of intersection is the common vertex. They alw^ays have the same 
number of degrees. 

12. A dihedral angle is one formed by the intersection of two 
surfaces at various degrees. 

13. Complementary angles are such that together they form a 
right angle, /. e,, add up to 90 degrees. 

14. Supplementary angles are such that together they form two 
right angles, i. e., add up to 180 degrees. 



42 



Notes on Drawing. 



CURVED SURFACES 



I 




Cylindrioal Surface 



Conloal Surface 



PLANE FIGURES 




Re^lar Polygon 





Irregular Polygon 




Triangle 



Pentagon 



Hexagon 




Octagon 



Geometrical Definitions. 43 

IV. Surfaces 

1. A surface has only two dimensions, length and breadth. 

2. A plane surface is one that is perfectly flat. If a straight edge 
be laid on it in any position the same will touch the surface at 
every point. 

3. A curved surface is one that is not a plane surface. 

a. A cylindrical surface is a curved surface generated by a 
straight line which constantly intersects a given curve and 
remains parallel to itself. 

b. A conical surface is a curved surface generated by a 
straight line which constantly intersects a given curve and 
passes through a fixed point, called the apex. 

V. Plane Figures 

1. A plane -figure is any part of a plane surface bounded by 
straight or curved lines. 

2. A polygon is a plane figure bounded by straight lines. 

a. A regular polygon is one where the bounding straight lines 
are all of the same length and the angles are all equal. 

b. An irregular polygon is one where the bounding straight 
lines are of different lengths. 

c. The sides are the straight lines bounding the polygon. 

d. The perimeter is the length of all the sides added together^ 
which is the whole distance around the figure. 

e. The angles of a polygon are the angles formed with each 
other by the bounding sides. 

/. The base of a polygon is the side upon which it rests. 
g. The altitude of a polygon is the length of the straight line 
drawn from the highest point perpendicular to the base, or 
the base extended. 
h. The diagonal is a line which joins two vertices and does 

not form a side. 
i. Regular polygons are named by the number of sides as 
follows : 
Triangle — 3 sides. Heptagon — 7 sides. 

Quadrilateral — 4 sides. Octagon — 8 sides. 

Pentagon — 5 sides. Decagon — 10 sides. 

Hexagon — 6 sides. Dodecagon — 12 sides. 



44 



Notes on Drawing. 



TRIANGLES 




iBOsoeles 





Equilateral 



Oblique 




c Similar or Proportional 




QUADRILATERALS 



Rectangle 



Square 



Rhomboid 



Trapezoid 




Trapezium 



Geometrical Definitions. 45 

3. A triangle is a polygon having three sides. 

a. An isosceles triangle is one having two of its sides of 
equal length. 

b. An equilateral- triangle is one in which all three sides are 
of equal length, which makes all the angles equal. 

c. A scalene triangle is one in which all of the sides are of 
different lengths.' 

d. A right-angled triangle is one in which one of the angles 
is a right angle, or has 90 degrees. The hypotenuse is 
the side opposite the right angle. 

e. An oblique triangle is one wdiich has no right angle and 
in which no two angles are equal. 

/. Equal triangles are such in which the sides of one are of 
the same length as the corresponding sides of the other. 

g. Similar triangles are such in which the angles of one are 
the same as the corresponding angles of the other. 

h. Proportional triangles are similar triangles. If the side of 
triangle A is twnce or any number of times the length of 
the corresponding side of similar triangle B, then all of the 
sides of triangle B are the same number of times the length 
of the corresponding sides of A. 

4. A quadrilateral is a plane figure having four sides. 

a. A parallelogram is a quadrilateral whose opposite sides are 
parallel. 

1. A rectangle is a parallelogram whose angles are all 
right angles. 

2. A square is a rectangle whose sides are equal. 

3. A rhomboid is a parallelogram whose angles are not 
right angles, but w^hose opposite sides only are equal. 

4. A rhombus is a parallelogram whose angles are not 
right angles, but all whose sides are equal. 

h. A trapezoid is a quadrilateral w^hich has only two of its 
sides parallel. 

c. A trapezium is a quadrilateral having no two sides parallel. 



45 



Notes on Drawing. 



CIRCLES 



^^3^itS£££e^^^ 




Concentric Circles 



Tangent Circles 



ELLIPSE 




ao-H3b=ad+db 



Geometrical Definitions. 47 

5. A circle is a plane figure boundecl by a curved line equi-distant 
at every point from the center. 

a. The diameter is any straight line drawn through the center 

and terminated by the curve on each side. All diameters 

are of equal length. 
h. The radius is any Straight line drawn from the center and 

terminated by the circumference. It is one-half the length 

of the diameter. 

c. The circumference is the curved line bounding the circle. 
Its length is called the perinieter. 

d. An arc is any part of the circumference. 

e. A chord is a straight line joining any tw^o points on the 
circumference. 

f. A segment is the area of space bounded by an arc and the 
chord joining its extremities. 

g. A sector is the area or space bounded by an arc and two 
radii connecting its extremities with the center. 

h. A quadrant is a sector where the radii are at right angles 
to each other. It is one-fourth of a circle. 

/ A tangent is a line which touches the circle at just one point 
of the circumference; it is ahvays perpendicular to a radius 
drawn to that point. Two circles are tangent to each 
when they touch at just one point; a line connecting the 
centers passes through the point of contact. 

y. Concentric circles are circles drawn from the same center. 

k. A semi-circle is one-half of a full circle. 

6. The ellipse is a plane figure bounded by a curved line such 
that the sum of the distances of any point on it from the two fixed 
points, called the foci, is always the same. 

a. A diameter is any line drawn through the center and ter- 
minated by the ellipse. 

h. The major axis is the longest diameter. The foci are 
located on it. 

c. The minor axis is the shortest diameter. 

d. The major and minor axis are always perpendicular to 
each other. 

e. Any diameter is divided into two equal parts by the center. 



48 



Notes on Drawing 



SOLID FIGURES 



^^^ 



I 



Faqe 



Base 



J 



>^ 



-k 




Cube 



Prlem 



Prlerc 







P 


•H 


4^ 

(D 

s 




r^ 


1 


-- -. 




Cylinder 



Pyramid 



Geomp:trical Dki-ixittoxs. 49 

VI. Solid Figures 

1. A solid figure is a figure which has three dimensions, length, 
breadth and thickness. 

2. A polyhedron is a solid bounded l)y plane surfaces. 
a. The faces are the sides which enclose the figure. 

b. The edges are the intersections of the faces. 

c. The entire surface is the area of all the surfaces enclos- 
ing it. 

d. The altitude is the height of the highest point, measured 
perpendicular to the base. 

e. A dihedral angle is an angle formed b}^ the intersection of 
any two faces. 

/. A solid angle is one formed by more than two intersecting 

planes passing through a common point called the apex, 
g. The volume is the space enclosed by the faces. 

2. A prism is a polyhedron whose sides are parallelograms and 
whose ends are equal polygons parallel to each other. 

a. A parallelopipedon is a prism whose bases are parallelo- 
grams. 

h. A cube is a parallelopipedon whose faces and bases are 
squares. 

.3 A cylinder is a closed cylindrical surface intercepted by tw^o 
planes wdiich form the bases. 

a. The axis of a cylinder is the line connecting the centers of 
the curves forming the bases. 

b. The altitude is the perpendicular distance between the two 
ends. 

c. An element is a straignt line on the surface which coincides 
wnth any one of the positions of the generating line. 

d. A right cylinder is one in which the axis is perpendicular 
to the base. 

4. A pyramid is a solid whose base is a polygon and whose sides 
are triangles uniting in a common point, called the vertex. 

a. The altitude of a pyramid is the height of the vertex 
measured perpendicular to the base. 

b. The slant height of a pyramid is the length of a line 
drawn from the vertex perpendicular to the sides of the 
base when these lines are all equal. 



50 



Notes on Drawing. 



5. The cojic is a closed conical surface intercepted by a plane. 

a. The altitude is the height of the vertex measured perpen- 
dicular to the base. 

b. The slant height is the distance from the vertex to the cir- 
cumference, measured perpendicular to latter. 

6. The frustum of a pyramid or cone is the part remaining after 
the top is cut ofif by a plane parallel to the base. 

7. The sphere is a solid generated by a semicircle revolved about 
the diameter. Any point on its surface is equi-distant from the 
center. 



SOLID FIGURES 




Cone 





Sphore 




Frustums 



44 



LIBRftRY OF CONGRESS 



019 945 442 m 



